Public Transport
Arriving at the airport I normally take the Metro, to the city. It leaves something like every 5
th minute. And 20 minutes later you are in the center. Being in Copenhagen you will find that bying tickets for the public transport is very expensive. But the public transport is very efficient, and you can buy a Copenhagen Card that gives you access to all public transport and to some tourist attractions. 24 or 72 hours.
In the S-train you can bring a standard bike for free.
And the Harbour buses go up and down the harbour and costs the same as a normal busticket.
If you are staying for a longer period, it is cheaper to get a Rejsekort
It is a plastic card, that can take you anywhere in the coutry, and it is cheaper, and you dont need to buy tickets. 

But the cheapest, fastest, healthy, environmental friendly way to go is off course by Bike. You can rent a bike here:
They also have guided tours. Read more here

Your own car
The trouble about driving a car is that it is very expensive to park it. In the city center it can cost as much as 30 kr – 4 Euro = 5 $ an hour. Then it is cheaper to park it outside the center. But be aware that the paying zone is going out 6-7 km from the center. The golden rule also counts here: the longer distance from the old city, the cheaper parking fees. Auto Camping areas are outside the city. , is really a dump and not cosy at all.

But there are many places you can just park your camper, without paying anything. Just remember to clean after yourself. 😉